In this section we shall define the complex number system as the set R × R (the Cartesian product of the set of reals, R, with itself) with suitable addition and multiplication operations. We shall define the real and imaginary parts of a complex number and compare the properties of the complex number system with those of the real number system, particularly from the point of view of analysis.

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In Section 4 we introduce the hyperbolic functions sinh, cosh and tanh, which are constructed from exponential functions. These hyperbolic functions share some of the properties of the trigonometric functions but, as you will see, their graphs are very different.

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. When prompted after exercise 2.2 to watch the video for this unit, return to this page and watch the four clips below. After you've watched the clips, return to the workbook.

Click 'View document' to open the workbook (PDF, 1.0 MB).

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The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. Once you have completed the workbook and exercises return to this page and watch the video below, ‘The arch never sleeps’, which discusses a practical application of some of the ideas in workbook.

Click 'View document' to open the workbook (PDF, 0.8

Author(s): The Open University

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School mathematics curricula often focus on lists of content objectives in areas like number, arithmetic, statistics, measurement, geometry, trigonometry, and algebra. A typical list of content objectives might contain over one hundred objectives to be introduced or revisited and learned each year. These can be seen as hierarchical in nature but many textbooks do not attempt to organise the objectives in ways that enable the bigger underpinning ideas to become apparent to the pupils. In addit

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One of the characteristics of ‘good’ information is that it should be balanced and present both sides of an argument or issue. This way the reader is left to weigh up the evidence and make a decision. In reality, we recognise that no information is truly objective.

This means that the onus is on you, the reader, to develop a critical awareness of the positions represented in what you read, and to take account of this when you interpret the information. In some cases, authors may be

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There is a variety of new approaches or terms that are interlinked, and have been prominent throughout this book. All of them have played a part in this book's journey through the scientific, political, philosophical and social implications of climate change.

Governance of climate change is about: decision making under uncertainty; understanding and representing vulnerability even when vulnerabilities are difficult to assess or unknowable; and making every aspect of human

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The proposers of step-by-step progress towards sustainability would include in their plans many of the ideas proposed in the previous two subsections. However, what distinguishes this group is that they stand in the middle of the scale between faith in unfettered business voluntarism and a conviction that radical transformations are required. Their incrementalism is reflected in the kinds of pragmatic solutions they propose; their radicalism shows in the way they think about new roles and pro

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It would be a serious error to imagine that ‘government’ has evaporated: it still shapes many aspects of our lives from beginning to end (welfare, taxation, transport and, of course, the recording of births and deaths). Governments are the central negotiators of environmental-change policies at international level, and of their implementation at national and local level. Nevertheless, for many areas of life, governance is undeniably a better description both of new processes that are alre

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Measuring the concentration of lead in the ice is called a direct measurement: the ice sample is melted and the water produced contains a very small but readily measured quantity of lead dust. A very accurate set of scales would be needed to measure it, but it is a directly measured quantity. There are also many indirect measurements that can be made using proxy data. The concept for using proxies is both simple and brilliant: one measured property allows inference about other states o

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Few people agree that individuals should take the main responsibility for tackling environmental issues. For example, in a 2007 poll of over 2000 UK citizens, 70% agreed that the government should take a lead in combating climate change, even if it means using the law to change people's behaviour. However, over 60% disagreed that there was nothing they could do to avert climate change and over half agreed that they would do more if others did more too, although 40% thought that recycling was

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You've been considering how to reduce your own carbon footprint to help tackle the worst effects of climate and other environmental changes. To that extent, ‘I’ as an individual consumer has a role to play.

But unless you live alone, you share your household with other people, a group that could be called ‘we’. Everyone in the household may have similar views on living lightly. But, even within a household, there may be different views and priorities about what, if anything, sho

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So far, you've been considering reductions in average individual or household carbon footprints by 20% to 30% or more.

But it is becoming increasingly clear that this will not be enough. As I mentioned in Section 4, developed countries, like Britain, Germany and America, will have to reduce their CO2e emissions by 60% to 80% or more by 2050 to prevent climate change running out of control, while at the same time allowing the growing populations of Africa, India and China to r

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Nobody made a greater mistake than he who did nothing because he could only do little.

(Edmund Burke, 1729–1797)

If you want to consider how to further lighten your carbon footprint, you may need more detailed information on the effect of technical and behavioural actions not covered by the carbon calculator, or included only as a part of other actions.

It's important to understand whic

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